MinT - Best Known (90−27, 90, s)-Nets in Base 4

Best Known (90−27, 90, s)-Nets in Base 4

(90−27, 90, 312)-Net over F₄ — Constructive and digital

Digital (63, 90, 312)-net over F₄, using

trace code for nets [i] based on digital (3, 30, 104)-net over F₆₄, using
- net from sequence [i] based on digital (3, 103)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 3 and N(F) ≥ 104, using
    - a function field by SÃ©mirat [i]

(90−27, 90, 440)-Net over F₄ — Digital

Digital (63, 90, 440)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−27, 90, 24997)-Net in Base 4 — Upper bound on s

There is no (63, 90, 24998)-net in base 4, because

1 times m-reduction [i] would yield (63, 89, 24998)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 383175 171813 185701 185589 478969 224603 068483 626691 047500 > 4⁸⁹ [i]